When you’re building a mathematical model, for any purpose (prediction, understanding, whatever), you have to keep it simple. You can’t literally model everything, any more than you can draw a map as big as the world itself. And if you want to get analytical results, as opposed to merely simulating the model, you have to keep it relatively simple.
There are two ways to keep things simple: leave stuff out, and leave stuff out while implicitly summarizing its effects.
The first strategy is the more obvious and familiar one. If you’re modeling competition, omit predators from your model. And if you don’t care about the effects of spatial or temporal abiotic environmental variation on competitive outcomes in your model, assume that the abiotic environment is constant in time and space. Etc.
The second strategy is just as common (indeed, both strategies are unavoidable), but in my experience readers don’t always recognize when it’s been used. You leave stuff out of your model, but implicitly summarize its effects on the stuff you’re modeling.
Indeed, pretty much any bit of any mathematical model can be thought of as implicitly summarizing the effects of some unspecified process(es). For instance, predator-prey models typically assume a constant conversion efficiency: a fixed number of predators (or units of predator biomass) produced per number or unit of prey consumed. But that conversion efficiency parameter actually summarizes a massive amount of unspecified underlying biology! All the digestive physiology and gut microbiota and gene expressiony…um, things and biochemical [mumble mumble] that go into turning food into predators just get black-boxed into one number.* Which is often fine! For many predators it really is the case that there’s a pretty fixed ratio of predators produced per prey consumed. Fully explaining why that ratio takes on the value it does in any particular case might require the work of many lifetimes–but who cares if all you’re trying to model is predator-prey dynamics?
In evolution, think of quantitative genetics and what Alan Grafen calls the “phenotypic gambit”. Don’t model the underlying genetic architecture of the phenotypic trait you’re modeling. Just assume the trait is the result of a normally-distributed genetic effect plus a normally-distributed environmental effect, plus a GxE interaction if you’re feeling fancy. It often works! For further examples, see this old post on how the concept of “genetic drift” (or in ecology, “demographic stochasticity”) is just a way of summarizing the consequences of lots of unspecified “low level” events, and this old post on trait-based ecology of phytoplankton.
High on the list of any theoretician’s pet peeves is when readers mix up these two strategies, thinking that something important has been omitted from the model when in fact its effects have been implicitly summarized. For instance, say you have a Tilman-esque resource competition model that obeys the R* rule. There’s no possibility of stable coexistence; whichever species has the lowest R* for the limiting resource outcompetes all the others. Now imagine that you add in intraspecific density dependence for the best competitor: make its per-capita growth rate decline as its own density increases (as well as remaining an increasing function of resource availability). The resulting model now will potentially allow stable coexistence of two species. But you have not thereby shown that two species can coexist without any niche differentiation. Because implicitly, you added in niche differentiation when you added in intraspecific density dependence. One of the species now competes intraspecifically for some unspecified reason that’s independent of competition for the shared resource. Which means that it must have some sort of “niche difference” from the other species. The effects of that unspecified niche difference are summarized by the intraspecific density dependence.
There’s a lot more that could be said here, in particular about the circumstances in which each of these two simplification strategies is best deployed. But I’ll leave that for others to say in the comments. 🙂
*I am not a physiologist, microbiologist, biochemist, or developmental biologist, in case you can’t tell.